0,3

a(n) = degree of Kac determinant at level n as polynomial in the conformal weight (called h). (Cf. C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Vol. 2, p. 533, eq.(98); reference p. 643, Cambridge University Press, (1989)) - comment from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that from any part z > 1 one can take an element of amount 1 in one way only. That means z is composed of z unlabeled parts of amount 1, i.e. z = 1 + 1 + ... + 1. E.g. for n=3 to n=2 we have A006218(3) = 6 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. For the case of z composed by labeled elements, z = 1_1 + 1_2 + ... + 1_z, refer to A066186. - Thomas Wieder (wieder.thomas(AT)t-online.de), May 20 2004

Number of times a derivative of any order (not 0 of course) appears when expanding the n-th derivative of 1/f(x). For instance (1/f(x))'' = (2 f'(x)^2-f(x) f''(x)) / f(x)^3 which makes a(2) = 3 (by counting k times the k-th power of a derivative). - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 07 2005

Starting with offset 1, = the partition triangle A008284 * [1, 2, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 13 2008

Erdos, Paul and Lehner, Joseph, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8, (1941). 335-345.

Kessler, I., and Livingston, M., The expected number of parts in a partition of n, Monatsh. Math. 81 (1976), no. 3, 203-212.

T. D. Noe, Table of n, a(n) for n=0..1000

G.f.: Sum n*x^n Product 1/(1-x^k); k = 1..n; n=1..inf.

G.f.: Sum x^k/(1-x^k); k=1..inf * Product 1/(1-x^m); m=1..inf.

a(n) = Sum of the number of divisors of m * partitions of n-m; m=1..n.

g:=sum(n*x^n*product(1/(1-x^k), k=1..n), n=1..60);

f[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; Table[ f[n], {n, 0, 41}]

CoefficientList[ Series[ Sum[n*x^n*Product[1/(1 - x^k), {k, n}], {n, 100}], {x, 0, 100}], x]

(PARI) f(n)= {local(v, i, k, s, t); v=vector(n, k, 0); v[n]=2; t=0; while(v[1]<n, i=2; while(v[i]==0, i++); v[i]--; s=sum(k=i, n, k*v[k]); while(i>1, i--; s+=i*(v[i]=(n-s)\i)); t+=sum(k=1, n, v[k])); t } (Thomas Baruchel (baruchel(AT)users(AT)sourceforge.net), Nov 07 2005)

a(n) = sum(k*A008284(n, k), k=1..n).

Cf. A093694, A066186, A000070.

Cf. A008284.

Sequence in context: A001975 A096220 A034333 this_sequence A079983 A028926 A038577

Adjacent sequences: A006125 A006126 A006127 this_sequence A006129 A006130 A006131

nonn,easy,nice

njas, clm